Nonparametric Bayes Classification and Hypothesis Testing on Manifolds.

Area of Science:

• Statistics
• Machine Learning
• Computational Geometry

Background:

• Predicting categorical variables with features on manifolds is challenging.
• Existing methods often struggle with complex, non-Euclidean data structures.
• Bayesian nonparametric approaches offer flexibility for modeling unknown distributions.

Purpose of the Study:

• To develop a general kernel mixture model for predicting categorical responses using manifold-structured features.
• To establish Bayesian nonparametric methods for testing distribution differences between groups on manifolds.
• To provide efficient computational algorithms for these statistical tasks.

Main Methods:

• A general kernel mixture model with a product kernel and an unknown mixing measure.
• Dirichlet process priors and von Mises-Fisher kernels for hypersphere manifolds.
• Bayesian methods including slice sampling for posterior computation.
• Bayes factor consistency for distribution comparison.

Main Results:

• Sufficient conditions for posterior consistency in distribution estimation are provided.
• Efficient Bayesian computation methods are developed for classification and testing.
• The proposed methods demonstrate effectiveness in simulation and spherical data applications.

Conclusions:

• The developed Bayesian nonparametric framework effectively handles categorical prediction and distribution testing on manifolds.
• The methods offer theoretical guarantees of consistency and practical computational efficiency.
• This work advances statistical modeling for complex, non-Euclidean data.